Annual Foundation Schools (AFS)

Objectives of AFS

Basic knowledge in algebra, analysis, discrete mathematics and topology forms the core of all advanced instructional schools the schools to be organized in this programme.
The objective of the Annual Foundation Schools, to be offered in Winter and Summer every year, is two fold:
  1. To bring up students with diverse backgrounds to a common level.
  2. To identify those who are fit for further training.
Any student who wishes to attend the advanced instructional schools is strongly encouraged to enroll in the Annual Foundation Schools.

Format of AFS

The topics listed in the syllabi will be quickly covered in the lectures. There will be intensive problem sessions in the afternoons. The objective will not be to cover the syllabus prescribed, but to inculcate the habit of problem solving. However, the participants will be asked to study all the topics in the syllabus at home since the syllabi of these schools will be assumed in all the advanced instructional schools devoted to individual subjects.

Participants in AFS

These schools will admit 40 students in their first and second years of Ph. D. programme, students of M. Sc. (II Year), university lecturers and college teachers who lack the knowledge of basic topics covered in these schools.

A participant who has attended AFS-I and II will never be allowed to attend these again.

Syllabus for the Annual Foundation School (AFS)-I (Dec., 2005)

Algebra


RAVI A. RAO
Modules over PIDs:
Basic theory, applications to abelian groups and canonical forms.(6 lectures)
Introduction to Field Theory: Splitting fields, Separable and normal extensions (2 lectures)
J. K. VERMA
Algebraically closed fields. (1)
Fundamental theorem of Galois theory with applications to fundamental theorem of algebra and constructibility of regular polygons. (2)
Galois groups of cubics and quartics Cyclotomic extensions with some number theory applications (3)
Galois's solvability criterion, existence of Galois extensions of Q with given abelian group (2)
S. A. KATRE
Finite fields with applications to quadratic reciprocity.(2 lectures)
Norms and traces, Hilbert's theorem 90, Artin-Schreier theorem with number theory applications (3 lectures)
Luroth's theorem with applications to algebraic curves. Galois group of K(T)/K (3 lectures)
Texts/References:
M. Artin, Algebra, Prentice-Hall of India, New Delhi, 1994. D. S. Dummit and R. M. Foote, Abstract Algebra, 2nd Edn., John Wiley & Sons, Inc., New York (Asian Edn., Singapore), 2003. N. Jacobson, Basic Algebra, Vol. 1, Freeman & Co., USA, Hindustan Publishing Corporation (India), Delhi, Reprint, 1991. S. Lang, Algebra, 3rd edn., Addison Wesley Pub. Co., Inc., USA, 1993. I.S. Luthar, I.B.S. Passi, Algebra-IV, Field Theory, Narosa, 2004.

Analysis


R. R. SIMHA
Review of the Riemann integral (1 lecture)
Construction of the Lebesgue measure on R n (2 lectures) Abstract Integration Theory and the main convergence theorems (2 lectures)
The L p spaces and Applications (3 lectures)
Pre-requisites for Measure and Integration:
Standard properties of real numbers including lim sup and lim inf of sequences. Topology of metric spaces. Compact metric spaces. Complete metric spaces. Baire's Theorem. Uniform convergence. Elementary properties of Riemann integral.
References :
W. Rudin: Real and Complex Analysis. (This book contains all the basic material and many applications; the exercises are a very valuable part of the book.)
Saks: Theory of the Integral. (This book contains a complete and brief exposition of the abstract theory of Lebesgue integration.)
F. Riesz and B. S. Nagy: Functional Analysis. (This book is written in a leisurely style, and contains a wealth of information. Very good for browsing.)
E. H. Lieb and M. Loss - Analysis. (This book is written for analysts and physicists, and contains much non-standard material.)
A. S. Athavale
Fourier Series and Functional Analysis
1. Conditional, unconditional and absolute convergence of a series in a normed linear space; notion of an orthonormal basis for a Hilbert space (1 lecture)
2. Trigonometric series, Fourier series, Fourier sine and cosine series (1 lecture)
3. Piecewise continuous/smooth functions, absolutely continuous functions, functions of bounded variation (and their significance in the theory of Fourier series) (1 lecture)
4. Generalised Riemann-Lebesgue lemma (1 lecture)
5. Dirichlet and Fourier kernels (2 lectures)
6. Convergence of Fourier series (1 lecture)
7. Discussion (without proofs) of some of the following topics: The Gibbs phenomenon, divergent Fourier series, term-by-term operations on Fourier series, various kinds of summability, Fejer theory, multivariable Fourier series (1 lecture).

Depending upon the feedback from students, the above syllabus is subject to minor (but not major) modifications. A prerequisite for the course is a sound knowledge of Calculus and Riemann Integration Theory. Some familiarity with Lebesgue Integration Theory and elementary Hilbert Space Theory is desirable, but (hopefully) not absolutely essential; in any case, the results used from those theories will be stated explicitly.
References :
1) George Bachman, Lawrence Narici and Edward Beckenstein, Fourier and Wavelet Analysis, Springer-Verlag, New York, 2000.
2) Richard L. Wheeden and Antoni Zygmund, Measure and Integral, Marcel Dekker Inc., New York, 1977.
3) Balmohan V. Limaye, Functional Analysis, New Age International (P) Ltd., New Delhi, 2004.
H. Bhate
Basic theory of ordinary differential equations:
Existence of local solutions for first order systems, maximal time of existence, finite time blow-up, global solutions.
Gronwall inequality, Continuous dependence on initial data and on the vector field on bounded intervals.
Examples of linear systems, Fundamental solutions. (8 lectures)

Texts/References:
1)  Differential Equations, Dynamical Systems and an Introduction to Chaos,
     2nd Edn., by M. W. Hirsch, S. Smale, R. L. Devaney, Elsevier, 2004.
2)  Real Analysis by G. B. Folland, John Wiley, 1999.
3)  Real And Complex Analysis by W. Rudin, McGraw Hill, 1987.

Differential Geometry and Topology

R. V. Gurjar
(1) Smooth maps, bump functions, smooth partitions of unity. Inverse and implicit function theorem.
Examples. (3 lectures)
(2) Manifolds, tangent space, immersion and submersion, regular and critical values, Sard's theorem.
Applications. (4 lectures)
Ravi Kulkarni
(3) Integration of forms, Stokes' theorem  (3 lectures)
(4) Classification of 2-manifolds. (4 lectures)
(5) Differential geometry of curves in R2, R3 and surfaces in R3.
(3 lectures)
G. Santhanam
(6) Introduction to Riemannian Geometry (7 lectures)
--------------------

(7) Morse Theory (optional topic)

Texts and references:
1) M. P. do Carmo, Differential Geometry of Curves and Surfaces, Prentice Hall, Engelwood, NJ 1976.
2) A. Gray, Modern Differential Geometry of Curves and Surfaces with MATHEMATICA, CRC Press, 1998.
3) J. Hicks, Notes on Differential Geometry.
4) S. Kumaresan, A Course on Differential Geometry and Lie Groups, Texts and Readings in Mathematics 22, Hindustan Book Agency, 2002.
5) S. Kumaresan, A Course in Riemannian Geometry, (Lecture Notes).
6) S. Kumaresan, Classification of Surfaces via Morse theory, Expositions Mathematica, 18 (2000), 37-74.
7) W. Massey, Algebraic Topology, GTM Series, Springer Verlag, 127
8) John Milnor, Morse Theory, Annals of Math. Studies, 51, Princeton.
9) M. Spivak, A Comprehensive Introduction to Differential Geometry, vol. I-V, Publish or Perish.
10) J. A. Thorpe: Elementary Topics in Differential Geometry, Undergraduate Texts in Mathematics, Springer 1979

(4,5,6,7,8,10 will be referred to in the lectures.
Other possible references for Differential Topology: Guillemin and Pollak, Hirsch, Kosinsky)

Special Lecture Series (UM Lectures)

1. Nitin Nitsure
`Evolution of spaces' (6 Lectures)
(This will be a historical introduction to geometry from Euclid to present times.)
2. Dinesh Thakur

Syllabus of the Annual Foundation School-I (May-June, 2004)

Algebra I
(1) Modules over PIDs: The basic theory, structure theorem for f.g. abelian groups and canonical forms of matrices. (6 lectures)
(2) Galois theory: Separable and normal extensions, algebraically closed fields, splitting fields, Fundamental theorem of Galois theory, Galois groups of cubic and quartics, fundamental theorem of algebra, finite fields, Galois's solvability criterion, cyclotomic and abelian extensions, (12 lectures)
(3) Representation theory of finite groups:Permutation representations, character theory and orthogonality relations, Burnside's theorem, representations of SU2. (6 lectures)
Texts/References:
1. N. Jacobson, Basic Algebra I.
2. S. Lang, Algebra, 3rd edition.
3. M. Artin, Algebra.
4. Dummit and Foote, Algebra.
Analysis
Real Analysis Basics: Measures, Integration, Normed spaces, Baire category theorem. Open mapping theorem, Closed graph theorem, Uniform boundedness theorem. (12 lectures)
Introduction to Fourier Analysis:(6 lectures) Basic theory of ordinary differential equations: Existence of local solutions for first order systems, maximal time of existence, finite time blow-up, global solutions. Gronwall inequality, Continuous dependence on initial data and on the vector field on bounded intervals. Examples of linear systems, Fundamental solutions. (6 lectures)
Texts/References:
1. Real Analysis by G.B.Folland, John Wiley, 1999.
2. Real And Complex Analysis by W. Rudin, McGraw Hill, 1987.
Differential geometry and topology
(1) Smooth maps, bump functions, smooth partition of unity. Inverse and implicit function theorems.(4 lectures)
(2) Manifolds, tangent space, immersions submersions. Regular and critical values, Sard's theorem. (5 lectures)
(3) Transversality. Embedding manifolds in euclidean spaces. Classification of 1-dim. manifolds. Orientability. (5 lectures)
(4) Intersection theory and applications: Normal bundle and epsilon-nbds; Brouwer's degree of a map, winding number. Brouwer's fixed point theorem, Fundamental Theorem of Algebra, Jordan-Brouwer's separation theorem. (6 lectures)
(5) Vector fields, Poincare-Hopf index theorem, Hopf degree theorem. (4 lectures)3
References/Texts:
1. V. Guillemin, and A. Pollack, Differential Topology.
2. A. A. Kosinski, Differential Manifolds. 138, Pure and applied Mathematics, Academic Press.
3. John Milnor, Topology from the differentiable viewpoint, Univ. Press of Virginia, Charlottesville, USA, 1965.
Combinatorics
(1) Graph Theory: Connectivity, network flows, matchings, planarity and duality, matrix tree theorem, spectra of graphs, graph colorings, Ramsey theory.
(2) Enumerative Combinatorics: Basic counting coefficients, generating functions, principle of inclusion and exclusion, partitions, exponential formula, Lagrange inversion formula, symmetric functions, Polya theory, posets and mobius inversion.
Text/References:
1. B. Bollobas, Modern Graph Theory, Springer-Verlag, GTM.
2. D. B. West, Graph Theory, Prentice Hall of India.
3. Diestel, Graph Theory, Springer-Verlag, GTM.
4. J. H. van Lint & R. M. Wilson,A course in Combinatorics, Cambridge.
5. R. P. Stanley, Enumerative Combinatorics, Cambridge.

Syllabus for Annual Foundation School-II (December 2004)

Algebra II
(1) Homological algebra: Derived functors, projective modules, injective modules, free and projective resolutions, tensor, exterior and symmetric algebras, injective resolutions, Ext and Tor. ( 12 lectures)
(2) Basic commutative algebra: Prime ideals and maximal ideals, Zariski topology, Nil and Jacobson radicals, Localization of rings and modules, Noetherian rings, Hilbert Basis theorem, modules, primary decomposition, integral dependence, Noether normalization lemma, principal ideal theorem, Hilbert's Nullstellensatz, structure of artinian rings, Dedekind domains. (12 lectures)
Text/References:
1. M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra.
2. D. Eisenbud, Commutative algebra with a view towards algebraic geometry.
Complex analysis
(1) Analytic functions, Path integrals, Winding number, Cauchy integral formula and consequences. Hadamard gap theorem, Isolated singularities, Residue theorem, Liouville theorem.4
(2) Casorati-Weierstrass theorem, Bloch-Landau theorem, Picard's theorems, Mobius transformations, Schwartz lemma, Extremal metrics, Riemann mapping theorem, Argument principle, Rouche's theorem.
(3) Runge's theorem, Infinite products, Weierstrass p-function, Mittag-Leffler expansion.
Text: Complex analysis by Murali Rao & H. Stetkaer, World Scientific, 1991
Algebraic topology
(1) Basic notion of homotopy; contractibility, deformation etc. Some basic constructions such as cone, suspension, mapping cylinder etc. fundamental group; computation for the circle. Covering spaces and fundamental group. Simplicial Complexes, CW complexes.
(2) Simplicial Complexes, CW complexes. Homology theory and applications: Simplicial homology, Singular homology, Cellular homology of CW-complexes, Jordan-Brouwer separation theorem, invariance of domain, Lefschetz fixed point theorem etc.
(3) Categories and functors; Axiomatic homology theory.
Texts/References
1. E. H. Spanier, Algebraic Topology, Tata-McGraw-Hill
2. A. Hatcher, Algebraic topology, Cambridge University Press.
Number Theory Arithmetic functions, congruences, quadratic residues, quadratic forms, Diophantine approximations, quadratic fields, Diophantine equations.
Text/References :
1. A. Baker, Theory of numbers.
2. K. Ireland and M. Rosen, A classical introduction to modern number theory.